MCQ
The angle between the two diagonals of a cube is:
- ✓$30^\circ$
- B$45^\circ$
- C$\cos^{-1}\Big(\frac{1}{\sqrt{3}}\Big)$
- D$\cos^{-1}\Big(\frac{1}{{3}}\Big)$
✓
Answer
Correct option: A.
$30^\circ$
Let a be the length of an edge of the cube and let one corner be at the origin as shown in the figure. Clearly, $OP, AR,$ Consider the diagonals $OP$ and $AR.$
Direction ratios of $OP$ and $AR$ are proportional to $a - 0, a - 0, a - 0$ and $0 - a, a - 0, a - 0, e.i. a, a, a$ and $-a, a, a,$ respectivelly.
Let $\theta$ be the angle between $OP$ and $AR.$ Then,
$\cos\theta=\frac{\text{a}\times-\text{a}+\text{a}\times\text{a}+\text{a}\times\text{a}}{\sqrt{\text{a}^2+\text{a}^2+\text{a}^2}\sqrt{(-\text{a})^2+\text{a}^2+\text{a}^2}}$
$\Rightarrow\cos\theta=\frac{-\text{a}+\text{a}^2+\text{a}^2}{\sqrt{3\text{a}^2}\sqrt{3\text{a}^2}}$
$\Rightarrow\cos\theta=\frac{1}{3}$
$\Rightarrow\theta=\cos^{-1}\Big(\frac{1}{3}\Big)$
Similarly, the angles between other pairs of the diagonals are equal to $\cos^{-1}\Big(\frac{1}{3}\Big)$ as the angle between any two diagonals.
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